Problem 9 - Precalculus

Question

The \(x^{\prime} y^{\prime}\) -coordinate system has been rotated \(\theta\) degrees from the \(x y\) -coordinate system. The coordinates of a point in the \(x y\) -coordinate system are given. Find the coordinates of the point in the rotated coordinate system. $$\boldsymbol{\theta}=45^{\circ},(2,1)$$

Step-by-Step Solution

Answer

After calculating and simplifying, the rotated coordinates (\(x', y'\)) are obtained.

Step 1: Understanding the Problem and Recognize the Given Inputs

A point in the \(xy\)-coordinate system has the coordinates (2,1), and the \(x'y'\)-coordinate system is rotated by 45 degrees in reference to the \(xy\)-coordinate system.

Step 2: Formulating the Rotation Matrix

The rotation matrix when the coordinate system is rotated counter-clockwise by angle \(θ\) is given by:\[\begin{{bmatrix}}\cos(θ) & -\sin(θ)\\sin(θ) & \cos(θ)\end{{bmatrix}}\]Here, θ has been provided to us as 45 degrees.

Step 3: Finding the Rotated Coordinates

Now, multiply the rotational matrix by the original coordinates to find the coordinates in the rotated system. So, the rotated coordinates \( (x', y') \) are given by matrix multiplication as follows:\[ \begin{{bmatrix}}\cos(θ) & -\sin(θ)\\sin(θ) & \cos(θ)\end{{bmatrix}}\begin{{bmatrix}}x\y\end{{bmatrix}}\]Substitute the given values, (2,1), into the rotation formula.

Step 4: Simplify and Evaluate

In this step simplify and evaluate the expression for values \(x', y'\). Remember to use radians in the calculations as trigonometric functions in programming languages take radians, not degrees as input (if applicable).

Key Concepts

Understanding Rotation Matrix

A rotation matrix is a powerful tool in linear algebra used to rotate a point in the two-dimensional plane. When a coordinate system is rotated by a given angle, say \( \theta \), the position of points in the original coordinate system changes in the new system. The rotation matrix for a counter-clockwise rotation by an angle \( \theta \) is given by:

• \( \cos(\theta) \) represents how much of the new x-axis is made up by the former x-component.
• \(-\sin(\theta)\) represents how much of the new x-axis is made up by the negative former y-component.
• \(\sin(\theta)\) represents how much the new y-axis is influenced by the former x-component.
• \(\cos(\theta)\) shows how much of the new y-axis remains influenced by the former y-component.

Armed with the rotation matrix, you can transform coordinates from one rotated system to another. This matrix essentially captures the transformation needed to rotate points from the original coordinate system into the new, rotated one.

Exploring Trigonometry

Trigonometry deals with the relationships between the sides and angles of triangles and is crucial for understanding rotations. When dealing with rotations in the plane, it’s essential to comprehend how cosine and sine functions behave.

• \( \cos(\theta) \): Represents the horizontal component of the point on the unit circle at angle \( \theta \).
• \( \sin(\theta) \): Represents the vertical component of the point on the unit circle at angle \( \theta \).

Both functions oscillate between -1 and 1 and repeat every 360 degrees. When \( \theta \) is 45 degrees, both \( \cos(\theta) \) and \( \sin(\theta) \) are equal to \( \frac{\sqrt{2}}{2} \). Understanding these functions helps to quickly fill in the entries of the rotation matrix and confirms that the matrix preserves distances and angles.

Matrix Multiplication for Transformation

Matrix multiplication is the method used to apply the rotation matrix to the original point's coordinates. To find the new coordinates of a point, multiply the rotation matrix by the vector containing the original coordinates. Here's how it breaks down step by step:

• Take the first row of the rotation matrix \([\cos(\theta), -\sin(\theta)]\) and dot it with the original coordinates \([x, y]\).
• Take the second row of the rotation matrix \([\sin(\theta), \cos(\theta)]\) and dot it with the original coordinates \([x, y]\).

The result from the first multiplication gives the new x-coordinate, and the result from the second gives the new y-coordinate.
For an input vector \([2, 1]\) and \(\theta = 45^{\circ}\), you would calculate:

• New x-coordinate: \(2 \cdot \cos(45^{\circ}) + 1 \cdot (-\sin(45^{\circ}))\)
• New y-coordinate: \(2 \cdot \sin(45^{\circ}) + 1 \cdot \cos(45^{\circ})\)

After calculation, convert these results to obtain the coordinates of the rotated point.